Rotate and slant projector for fast fully-3d iterative tomographic reconstruction

ABSTRACT

Disclosed herein are embodiments of a rotate-and-slant projector that takes advantage of symmetries in the geometry to compute truly volumetric projections to multiple oblique sinograms in a computationally efficient manner. It is based upon the 2D rotation-based projector using the fast three-pass method of shears, and it conserves the 2D rotator computations for multiple projections to each oblique sinogram set. The projector is equally applicable to both conventional evenly-spaced projections and unevenly-spaced line-of-response (LOR) data (where the arc correction is modeled within the projector). The LOR-based version models the exact location of the direct and oblique LORs, and provides an ordinary Poisson reconstruction framework. Speed optimizations of various embodiments of the projector include advantageously utilizing data symmetries such as the vertical symmetry of the oblique projection process, a coarse-depth compression, and array indexing schemes which maximize serial memory access.

This invention was made with government support under Grant R01 CA107353awarded by the National Institutes of Health (NIH) and National CancerInstitute (NCI). The government may have certain rights in theinvention.

BACKGROUND

The present invention relates to the art of diagnostic imaging. Inparticular, it relates to positron emission tomography (PET) and otherdiagnostic modes in which a subject is examined and an image of thesubject is reconstructed from information obtained during theexamination.

Previously, PET has been used to study a radionuclide distribution insubjects. Typically, one or more radiopharmaceuticals are injected intoa subject. The radiopharmaceuticals are commonly injected into thesubject's blood stream for imaging the circulatory system or for imagingspecific organs which absorb the injected radiopharmaceuticals. Detectorelements are placed proximate to a surface of the subject to monitor andrecord emitted radiation. In some instances, the detector elements maybe rotated or indexed around the subject to monitor the emittedradiation from a plurality of directions, while in other instances aplurality of detector elements can be placed in fixed locations adjacentto the subject or a portion of the subject. For instance, detectorelements may be formed into rings and the rings placed substantiallyadjacent to one another to form a cylindrical scanner whereby a subjectis axially inserted at least partially into the cylinder of detectorelements. These are generally known as ring-type scanners. The monitoredradiation data from the multiplicity of directions is reconstructed intoa three dimensional image representation of the radiopharmaceuticaldistribution within the subject.

Iterative statistical reconstruction techniques in PET provide a robustframework for accounting for the statistical quality of the measureddata, modeling the image acquisition process, and incorporating priorknowledge (if any) about the reconstruction solution. While iterativereconstruction has generally become the standard for PET, completeutilization of such methods has been limited due to large computationaldemands. This is especially true for fully-3D PET, where both direct andoblique coincidence lines are measured to produce highly sensitive, butvery large datasets. A variety of approaches have been proposed foriterative reconstruction of fully-3D PET data. The most direct, andpotentially highest quality, implementation requires raw data beoperated upon directly by a reconstruction algorithm, thereby makingfull use of Poisson-based statistical models and avoiding anyunnecessary degradation or blurring accompanying data pre-processingsteps (e.g. arc-correction). However, these implementations tend to bethe most computationally demanding as well. At the other end of thespectrum, pre-processing methods, such as rebinning fully-3D data into aset of 2D sinograms followed by 2D iterative reconstruction, can be usedto greatly speed the reconstruction; however, such methods tend to spoilthe Poisson statistics of the data and/or introduce undesired blurringor other degradations.

Therefore, what is needed is a means to overcome challenges found in theart, some of which are described above.

SUMMARY

Described herein are embodiments of a projector (and backprojector) foriterative fully-3D PET reconstruction that offers full utilization ofiterative statistical reconstruction algorithms, has low computationalcost, and is amenable to modeling the physics and spatially-variantresolution effects of the PET acquisition. The projector (andbackprojector) can map to either regularly-spaced (“arc corrected”)projections or directly from the image to the raw coincidenceline-of-response (LOR) measurement space (and vice versa), such that theprojection operation explicitly models the non-uniform spacing of thecoincidence lines acquired by modern PET tomographs.

One aspect according to the present invention is a projector foriterative fully-3D reconstruction of an image by positron emissiontomography (PET) using a scanner comprised of a plurality of detectorelements. The projector comprises a volume image memory containing aresultant image volume that is initially estimated and stored in animage matrix. Further comprising the projector is a projection matrix.The projection matrix is comprised of planes of projection datacollected at each of a plurality of data collection directions. The datacollection directions include azimuthal angles of projection rays orlines-of-response that are perpendicular to the axis of symmetry of thescanner between detector elements in a same transaxial plane of thescanner and axial angles of oblique projection rays or lines of responsebetween detector elements in different transaxial planes of the scanner.The projector also includes a rotator that re-samples the resultantimage volume from the volume image memory by rotating the image matrixto a given azimuthal angle of the projection matrix and slanting thecolumns of the image matrix to each of the axial angles of obliqueprojection rays or lines-of-response at that given azimuthal angle suchthat columns of the image matrix align with each of the plurality ofdata collection directions of the projection matrix. The rotator appliesa coarse-depth compression factor to the image matrix such that rows ofthe image matrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed and the image is projected by summing thecompressed columns of the image matrix.

Another aspect according to the present invention is a projector foriterative fully-3D reconstruction of an image by positron emissiontomography (PET) using a scanner comprised of a plurality of detectorelements. The projector comprises a volume image memory containing aresultant image volume that is initially estimated and stored in animage matrix. Further comprising the projector is a projection matrix.The projection matrix is comprised of planes of projection datacollected at each of a plurality of data collection directions. The datacollection directions include azimuthal angles of projection rays orlines-of-response that are perpendicular to the axis of symmetry of thescanner between detector elements in a same transaxial plane of thescanner and axial angles of oblique projection rays or lines of responsebetween detector elements in different transaxial planes of the scanner.The projector also includes a rotator that re-samples the resultantimage volume from the volume image memory by rotating the image matrixto a given azimuthal angle of the projection matrix and slanting thecolumns of the image matrix to each of the axial angles of obliqueprojection rays or lines-of-response at that given azimuthal angle suchthat columns of the image matrix align with each of the plurality ofdata collection directions of the projection matrix. The image rotationand slanting is performed using volume-of-overlap between 3D volumes ofimage matrix voxels and a 3D volume of said projection rays orlines-of-response and the oblique projections rays or lines-of-response,such that projection by summing the columns of the image matrixintegrates the 3D volume contained within said volume-of-overlap.

Another aspect according to an embodiment of the present invention is aprojector for iterative fully-3D reconstruction of an image by positronemission tomography (PET) using a scanner comprised of a plurality ofdetector elements. The projector comprises a volume image memorycontaining a resultant image volume that is initially estimated andstored in an image matrix. Further comprising the projector is aprojection matrix. The projection matrix is comprised of planes ofprojection data collected at each of a plurality of data collectiondirections. The data collection directions include azimuthal angles ofprojection rays or lines-of-response that are perpendicular to the axisof symmetry of the scanner between detector elements in a sametransaxial plane of the scanner and axial angles of oblique projectionrays or lines of response between detector elements in differenttransaxial planes of the scanner. The projector also includes a rotatorthat re-samples the resultant image volume from the volume image memoryby rotating the image matrix to a given azimuthal angle of theprojection matrix and slanting the columns of the image matrix to eachof the axial angles of oblique projection rays or lines-of-response atthat given azimuthal angle such that columns of the image matrix alignwith each of the plurality of data collection directions of theprojection matrix. The rotator applies a coarse-depth compression factorto the image matrix such that rows of the image matrix are collapsedonto one another in accordance with the coarse-depth compression factorso depth information of the rotated image matrix is compressed.Furthermore, image matrix rotation and slanting is performed usingvolume-of-overlap between 3D volumes of image matrix voxels and a 3Dvolume of said projection rays or lines-of-response and the obliqueprojection rays and lines-of-response, such that projection by summingthe columns of the image matrix integrates the 3D volume containedwithin the volume-of-overlap.

Yet another aspect of an embodiment according to the present inventionis a method of iterative fully-3D reconstruction of an image by positronemission tomography (PET) by a scanner having a plurality of detectorelements. The method comprises estimating a resultant image volume andstoring the image volume in an image matrix. The image matrix is rotatedinto alignment with an azimuthal angle of a projection ray orline-of-response between detector elements in a same transaxial plane ofthe scanner. The projection ray or line-of-response is perpendicular toan axis of symmetry of the scanner. The image volume is re-sampled atthe azimuthal angle by slanting the image matrix to each axial angle ofoblique projection rays or lines of response at that azimuthal angle.The oblique projection ray or line-of-response is between detectorelements in different transaxial planes of the scanner such that columnsof the image matrix align with a plurality of data collectiondirections. A coarse-depth compression factor is applied to the imagematrix such that rows of the image matrix are collapsed onto one anotherin accordance with the coarse-depth compression factor so depthinformation of the rotated image matrix is compressed. The image isprojected by summing the compressed columns of the image matrix.

Another aspect of an embodiment according to the present invention is amethod of iterative fully-3D reconstruction of an image by positronemission tomography (PET) by a scanner having a plurality of detectorelements. The method comprises estimating a resultant image volume andstoring the image volume in an image matrix. The image matrix is rotatedinto alignment with an azimuthal angle of a projection ray orline-of-response between detector elements in a same transaxial plane ofthe scanner. The projection ray or line-of-response is perpendicular toan axis of symmetry of the scanner. The image volume is re-sampled atthe azimuthal angle by slanting the image matrix to each axial angle ofoblique projection rays or lines of response at that azimuthal angle.The oblique projection ray or line-of-response is between detectorelements in different transaxial planes of the scanner such that columnsof the image matrix align with a plurality of data collectiondirections. The image is projected by summing the columns of the imagematrix by integrating a 3D volume contained within a volume-of-overlap.The image matrix rotation and slanting is performed using thevolume-of-overlap between 3D volumes of image matrix voxels and a 3Dvolume of the projection rays or lines-of-response and the obliqueprojection rays or lines-of-response.

Another aspect of an embodiment according to the present invention is amethod of iterative fully-3D reconstruction of an image by positronemission tomography (PET) by a scanner having a plurality of detectorelements. The method comprises estimating a resultant image volume andstoring the image volume in an image matrix. The image matrix is rotatedinto alignment with an azimuthal angle of a projection ray orline-of-response between detector elements in a same transaxial plane ofthe scanner. The projection ray or line-of-response is perpendicular toan axis of symmetry of the scanner. The image volume is re-sampled atthe azimuthal angle by slanting the image matrix to each axial angle ofoblique projection rays or lines of response at that azimuthal angle.The oblique projection ray or line-of-response is between detectorelements in different transaxial planes of the scanner such that columnsof the image matrix align with a plurality of data collectiondirections. A coarse-depth compression factor is applied to the imagematrix such that rows of the image matrix are collapsed onto one anotherin accordance with the coarse-depth compression factor so depthinformation of the rotated image matrix is compressed. The image isprojected by summing the columns of the compressed image matrix byintegrating a 3D volume contained within a volume-of-overlap. The imagematrix rotation and slanting is performed using the volume-of-overlapbetween 3D volumes of image matrix voxels and a 3D volume of theprojection rays or lines-of-response and the oblique projection rays orlines-of-response.

Yet another aspect according to an embodiment of the present inventionis a positron emission tomography (PET) scanner. The PET scanner iscomprised of a plurality of detector elements and a rotate and slantprojector. The rotate and slant projector is comprised of a projectionmatrix, an image matrix and a rotator. The rotator re-samples aresultant image volume from a volume image memory by rotating the imagematrix to a given azimuthal angle of the projection matrix and slantingcolumns of the image matrix to each axial angle of oblique projectionrays or lines-of-response at that given azimuthal angle such thatcolumns of the image matrix align with each of the plurality of datacollection directions of the projection matrix. The rotator applies acoarse-depth compression factor to the image matrix such that rows ofthe image matrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed and the image is projected by summing thecompressed columns of the image matrix.

Another aspect according to an embodiment of the present invention is apositron emission tomography (PET) scanner. The PET scanner is comprisedof a plurality of detector elements and a rotate and slant projector.The rotate and slant projector is comprised of a projection matrix, animage matrix and a rotator. The rotator re-samples a resultant imagevolume from a volume image memory by rotating the image matrix to agiven azimuthal angle of the projection matrix and slanting columns ofthe image matrix to each axial angle of oblique projection rays orlines-of-response at that given azimuthal angle such that columns of theimage matrix align with each of the plurality of data collectiondirections of the projection matrix. The rotator performs image matrixrotation and slanting using volume-of-overlap between 3D volumes ofimage matrix voxels and a 3D volume of the projection rays orlines-of-response and the oblique projection rays or lines-of-response,such that projection by summing columns of the image matrix integratesthe 3D volume contained within the volume-of-overlap.

Another aspect according to an embodiment of the present invention is apositron emission tomography (PET) scanner. The PET scanner is comprisedof a plurality of detector elements and a rotate and slant projector.The rotate and slant projector is comprised of a projection matrix, animage matrix and a rotator. The rotator re-samples a resultant imagevolume from a volume image memory by rotating the image matrix to agiven azimuthal angle of the projection matrix and slanting columns ofthe image matrix to each axial angle of oblique projection rays orlines-of-response at that given azimuthal angle such that columns of theimage matrix align with each of the plurality of data collectiondirections of the projection matrix. The rotator applies a coarse-depthcompression factor to said image matrix such that rows of the imagematrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed. Image matrix rotation and slanting isperformed by the rotator using volume-of-overlap between 3D volumes ofimage matrix voxels and a 3D volume of said projection rays orlines-of-response and the oblique projection rays and lines-of-response,such that projection by summing the columns of the image matrixintegrates the 3D volume contained within the volume-of-overlap.

Additional advantages of the invention will be set forth in part in thedescription which follows or may be learned by practice of theinvention. The advantages of the invention will be realized and attainedby means of the elements and combinations particularly pointed out inthe appended claims. It is to be understood that both the foregoinggeneral description and the following detailed description are exemplaryand explanatory only and are not restrictive of the invention, asclaimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are not to scale and are incorporatedin and constitute a part of this specification, illustrate embodimentsaccording to the invention and together with the description, serve toexplain the principles of the invention:

FIG. 1 a is an illustration of an embodiment of a computing device thatcan be used to practice aspects of the present invention;

FIG. 1 b is an alternative embodiment of a processing system that may beused to practice aspects of the present invention;

FIG. 1 c is an elementary exemplary representation of an embodiment of aprojector according to the present invention;

FIG. 1 d is a more-detailed exemplary representation of an embodiment ofa projector according to the present invention;

FIG. 2 illustrates an exemplary geometric arrangement of detectorelements of a scanner that may be used to practice the presentinvention;

FIG. 3 illustrates an exemplary process of a three-pass method ofshears, which breaks down the 2D rotation process into a series of threecomputationally efficient 1D shears;

FIG. 4 illustrates three steps of a rotate and slant projector accordingto embodiments of the present invention;

FIG. 5 is an exemplary illustration showing the volumetric nature ofLORs or projection rays;

FIGS. 6 a and 6 b are exemplary illustrations of the effect of usingcoarse-depth compression factors upon projection time (FIG. 6 a) andobject resolution in the axial direction (FIG. 6 b);

FIG. 7 shows example images of the Deluxe Jaszczak Phantom,qualitatively demonstrating the image quality that can be obtained withan embodiment of the projector described herein and 3D LOR-OSEM;

FIGS. 8 a and 8 b show the full projection and iterative reconstructiontimes as a function of the number of oblique datasets included accordingto an example of an embodiment of the present invention;

FIG. 9 shows how the projection time for the rotate-and-slant projectorincreases as a function of transaxial (transverse plane) image matrixsize according to an example of an embodiment of the present invention;

FIGS. 10 a-10 d illustrate results of a quantitative analysis of imagequality for an embodiment according to the present invention where threeimage quality measures were studied for the Deluxe Jaszczak phantom; and

FIG. 11 is a flow chart illustrating an exemplary process for iterativefully-3D reconstruction of an image by positron emission tomography(PET) by a scanner having a plurality of detector elements.

DETAILED DESCRIPTION

Before the present methods and systems are disclosed and described, itis to be understood that this invention is not limited to specificsynthetic methods, specific components, or to particular compositions,as such may, of course, vary. It is also to be understood that theterminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting.

As used in the specification and the appended claims, the singular forms“a,” “an” and “the” include plural referents unless the context clearlydictates otherwise. Ranges may be expressed herein as from “about” oneparticular value, and/or to “about” another particular value. When sucha range is expressed, another embodiment includes from the oneparticular value and/or to the other particular value. Similarly, whenvalues are expressed as approximations, by use of the antecedent“about,” it will be understood that the particular value forms anotherembodiment. It will be further understood that the endpoints of each ofthe ranges are significant both in relation to the other endpoint, andindependently of the other endpoint.

“Optional” or “optionally” means that the subsequently described eventor circumstance may or may not occur, and that the description includesinstances where said event or circumstance occurs and instances where itdoes not.

The present invention may be understood more readily by reference to thefollowing detailed description of preferred embodiments according to theinvention and the Examples included therein and to the Figures and theirprevious and following description.

As will be appreciated by one skilled in the art, the preferredembodiment may be implemented as a method, a data processing system, ora computer program product. Accordingly, the preferred embodiment maytake the form of an entirely hardware embodiment, an entirely softwareembodiment, or an embodiment combining software and hardware aspects.Furthermore, implementations of the preferred embodiment may take theform of a computer program product on a computer-readable storage mediumhaving computer-readable program instructions (e.g., computer software)embodied in the storage medium. More particularly, implementations ofthe preferred embodiments may take the form of web-implemented computersoftware. Any suitable computer-readable storage medium may be utilizedincluding hard disks, CD-ROMs, optical storage devices, or magneticstorage devices.

The preferred embodiments according to the present invention aredescribed below with reference to block diagrams and flowchartillustrations of methods, apparatuses (i.e., systems) and computerprogram products according to an embodiment of the invention. It will beunderstood that each block of the block diagrams and flowchartillustrations, and combinations of blocks in the block diagrams andflowchart illustrations, respectively, can be implemented by computerprogram instructions. These computer program instructions may be loadedonto a general purpose computer, special purpose computer, or otherprogrammable data processing apparatus to produce a machine, such thatthe instructions which execute on the computer or other programmabledata processing apparatus create a means for implementing the functionsspecified in the flowchart block or blocks.

These computer program instructions may also be stored in acomputer-readable memory that can direct a computer or otherprogrammable data processing apparatus to function in a particularmanner, such that the instructions stored in the computer-readablememory produce an article of manufacture including computer-readableinstructions for implementing the function specified in the flowchartblock or blocks. The computer program instructions may also be loadedonto a computer or other programmable data processing apparatus to causea series of operational steps to be performed on the computer or otherprogrammable apparatus to produce a computer-implemented process suchthat the instructions that execute on the computer or other programmableapparatus provide steps for implementing the functions specified in theflowchart block or blocks.

Accordingly, blocks of the block diagrams and flowchart illustrationssupport combinations of means for performing the specified functions,combinations of steps for performing the specified functions and programinstruction means for performing the specified functions. It will alsobe understood that each block of the block diagrams and flowchartillustrations, and combinations of blocks in the block diagrams andflowchart illustrations, can be implemented by special purposehardware-based computer systems that perform the specified functions orsteps, or combinations of special purpose hardware and computerinstructions.

Computer or Computing Device

In the embodiments referenced herein, a “computer” or “computing device”may be referenced. Such computer may be, for example, a mainframe,desktop, notebook or laptop, a hand held device such as a dataacquisition and storage device, or it may be a processing deviceembodied within another apparatus such as, for example, a scanner usedfor tomography. In some instances the computer may be a “dumb” terminalused to access data or processors over a network. Turning to FIG. 1 a,one embodiment of a computing device is illustrated that can be used topractice aspects of the preferred embodiment. In FIG. 1 a, a processor1, such as a microprocessor, is used to execute software instructionsfor carrying out the defined steps. The processor 1 receives power froma power supply 17 that also provides power to the other components asnecessary. The processor 1 communicates using a data bus 5 that istypically 16 or 32 bits wide (e.g., in parallel). The data bus 5 is usedto convey data and program instructions, typically, between theprocessor and memory. In the present embodiment, memory can beconsidered primary memory 2 that is RAM or other forms which retain thecontents only during operation, or it may be non-volatile 3, such asROM, EPROM, EEPROM, FLASH, or other types of memory that retain thememory contents at all times. The memory could also be secondary memory4, such as disk storage, that stores large amount of data. In someembodiments, the disk storage may communicate with the processor usingan I/O bus 6 instead or a dedicated bus (not shown). The secondarymemory may be a floppy disk, hard disk, compact disk, DVD, or any othertype of mass storage type known to those skilled in the computer arts.

The processor 1 also communicates with various peripherals or externaldevices using an I/O bus 6. In the present embodiment, a peripheral I/Ocontroller 7 is used to provide standard interfaces, such as RS-232,RS422, DIN, USB, or other interfaces as appropriate to interface variousinput/output devices. Typical input/output devices include localprinters 18, a monitor 8, a keyboard 9, and a mouse 10 or other typicalpointing devices (e.g., rollerball, trackpad, joystick, etc.).

The processor 1 typically also communicates using a communications I/Ocontroller 11 with external communication networks, and may use avariety of interfaces such as data communication oriented protocols 12such as X.25, ISDN, DSL, cable modems, etc. The communicationscontroller 11 may also incorporate a modem (not shown) for interfacingand communicating with a standard telephone line 13. Finally, thecommunications I/O controller may incorporate an Ethernet interface 14for communicating over a LAN. Any of these interfaces may be used toaccess a wide area network such as the Internet, intranets, LANs, orother data communication facilities.

Finally, the processor 1 may communicate with a wireless interface 16that is operatively connected to an antenna 15 for communicatingwirelessly with another device, using for example, one of the IEEE802.11 protocols, 802.15.4 protocol, or a standard 3G wirelesstelecommunications protocols, such as CDMA2000 1x EV-DO, GPRS, W-CDMA,or other protocol.

An alternative embodiment of a processing system that may be used isshown in FIG. 1 b. In this embodiment, a distributed communication andprocessing architecture is shown involving a server 20 communicatingwith either a local client computer 26 a or a remote client computer 26b. The server 20 typically comprises a processor 21 that communicateswith a database 22, which can be viewed as a form of secondary memory,as well as primary memory 24. The processor also communicates withexternal devices using an I/O controller 23 that typically interfaceswith a LAN 25. The LAN may provide local connectivity to a networkedprinter 28 and the local client computer 26 a. These may be located inthe same facility as the server, though not necessarily in the sameroom. Communication with remote devices typically is accomplished byrouting data from the LAN 25 over a communications facility to a widearea network 27, such as the Internet. A remote client computer 26 b mayexecute a web browser, so that the remote client 26 b may interact withthe server as required by transmitted data through the wide area network27, over the LAN 25, and to the server 20.

Those skilled in the art of data networking will realize that many otheralternatives and architectures are possible and can be used to practicethe preferred embodiments. The embodiments illustrated in FIGS. 1 a and1 b can be modified in different ways and be within the scope of thepresent invention as claimed.

Overview

Described herein are embodiments of a projector (and backprojector) foriterative fully-3D PET reconstruction that offers full utilization ofiterative statistical reconstruction algorithms, has low computationalcost, and is amenable to modeling the physics and spatially-variantresolution effects of the PET acquisition. The projector (andbackprojector) can map to either regularly-spaced (“arc-corrected”)projections or directly from the image to the raw coincidenceline-of-response (LOR) measurement space (and vice versa), such that theprojection operation explicitly models the non-uniform spacing of thecoincidence lines acquired by modern PET tomographs. The term“LOR-based” reconstruction is used to set apart methods that directlyreconstruct the raw coincidence-LORs of the scanner from those thatoperate on arc-corrected sinograms. In effect, the “arc correction” isincorporated into the projector/backprojector of the reconstructionitself, avoiding interpolations and other degradations that occur whenpre-processing projection data prior to reconstruction.

“LOR” is used generally to refer to the geometric linkage between a pairof detector elements in coincidence. In increasing order of complexity,a LOR may be approximated/interpreted as a line between the centers ofdetector elements, as a 2D strip between elements within a slice, as a3D volumetric “tube” connecting the surface areas of two detectorelements, or as the spatially-distributed 3D sensitive volume of imagespace that has potential to give rise to true coincidence events in thepair of detector elements in question.

The broad class of LOR-based reconstruction methods includes bothalgorithms that reconstruct from raw LOR histograms as well as listmodealgorithms that operate on an event-by-event basis, as are known tothose of ordinary skill in the art. By reconstructing directly from theraw LOR measurements, the Poisson statistical nature of the PET data ispreserved, and the full benefits of maximum-likelihood (ML) statisticalreconstruction can be realized.

Embodiments of the projector described herein, which can generally bereferred to as the rotate-and-slant projector, take advantage of certainsymmetries in the fully-3D geometry to provide efficient projection tomultiple oblique sinogram sets. Embodiments according to the presentinvention re-sample the image via rotation so that the columns alignwith the projection bins at a given azimuthal angle and implements a LORsampling scheme, which provides volumetric integration over thegeometric LOR volumes, and that is readily configured for incorporationof spatially distributed sensitive volumes (i.e., the spatially variantpoint spread function). Embodiments of the projector provides aconvenient image-based framework for modeling effects such as thespatially-variant geometric point response function and scatter; easilyadapt to a wide variety of scanner geometries; have modest memoryrequirements; and can be used to project and/or reconstruct any voxelsize without the need for any special setup. Software used to implementembodiments according to the present invention can be incorporated intoa computing device of a scanner, such as a PET scanner, or be configuredto analyze data from a scanner on a separated computing device.Non-limiting exemplary scanners that can be used according toembodiments of the present invention include General Electric Advance™high-resolution BGO PET scanners (GE Healthcare, United Kingdom) andBiograph Sensation 16™ Hi-REZ Model LSO PET/CT scanner (Siemens MedicalSolutions USA, Inc., Malvern, Pa.), though other scanners arecontemplated within the scope of the invention.

FIG. 1 c is an elementary representation of an embodiment of a projectorin accordance with the present invention. In FIG. 1 c, data from ascanner is received by a projector 100, acted upon by the projector, andoutput as a projection of the subject or a portion of the subject underconsideration. FIG. 1 d is a more detailed illustration of componentsthat comprise an exemplary projector 100. These components include amemory 102 containing an image matrix 104 and a projection matrix 106,as both are described herein, and a processor 108 whereupon operates arotator 110, also described herein.

Rotate-and-Slant Projector

FIG. 2 illustrates an exemplary arrangement of detector elements of ascanner that may be used to practice the present invention. Inparticular, FIG. 2 shows the (s, φ, z, δ) coordinate system used toparameterize fully-3D LORs for a generic ring PET tomograph. Embodimentsaccording to the present invention employ projection computations thatcan be grouped and shared among LORs based on symmetries of themeasurement geometry of the system shown in FIG. 2. These grouped andshared computations result in embodiments of a computationally-efficientfully-3D projector according to the present invention. Similar to arotation-based projector for 2D tomography, as known to one of ordinaryskill in the art, embodiments of a rotate-and-slant projector work byre-sampling the image stored as an estimated volume image representationin a volume image memory so that the columns of an image matrix alignwith rays or lines-of-reconstruction of a projection matrix. Projectionis then accomplished by summing the columns of the re-sampled imagematrix. In 2D, this re-sampling amounts to rotating the image matrix toazimuthal angle φ. Embodiments according to the present invention use athree-pass method of shears, as exemplary illustrated in FIG. 3, forthis rotation, which breaks down the 2D rotation process into a seriesof three computationally efficient 1D shears.

For fully-3D PET, the 2D rotation-based projector can be used forprojecting to direct sinograms (ring difference δ=0). For obliquesinograms (δ≠0), the projection rays or lines-of-reconstruction are notperpendicular to the axis of symmetry of the scanner and lie at somepolar angle θ. Here, the image matrix is slanted (axial shear) so thatthe re-sampled columns of the image matrix align with the obliqueprojection rays or lines-of-reconstruction as shown in FIG. 4.

FIG. 4 illustrates three steps of a rotate and slant projector accordingto the present invention. Embodiments of the projector include arotation processor or algorithm (also referred to as a “rotator”), whichrotates the estimated volume image representation in a volume imagememory to a selected projection direction. In Step 1, the image matrixis rotated by the rotator to azimuthal angle φ (refer to FIG. 2). AtStep 2, the columns of the image matrix are slanted (axially sheared) tothe ring difference, 6. This axial slant is a 1D depth-dependentinterpolation operation that can be performed much faster than the 2Drotation operation of Step 1. Furthermore, since the image can berotated once, stored in memory, then slanted repeatedly to all ringdifferences, the computational cost of projecting to multiple ringdifferences is a relatively small increase over projecting to a singlering difference. As a result, the rotate-and-slant projector has highcomputational efficiency for projecting to datasets with large numbersof ring differences. At Step 3 of FIG. 4, the columns of the re-sampledmatrix are summed as projection occurs down the columns. An integratoror summing means adds corresponding pixels or voxels of each of thelayers of data from the columns of the image matrix. The weighted sum ofthe layers forms a projection or reprojection of the estimated volumeimage representation in the volume image memory along the selecteddirection, in particular, the direction of projection rays orlines-of-reconstruction. A reconstruction algorithm forms a ratio ofcollected projection data and the reprojected projection. The ratio isbackprojected. The backprojected volumes of correction factors aresummed and used to correct the estimated volumetric image in thevolumetric image in the volumetric image memory. The corrected estimatedvolume image is again projected and the resultant reprojections areagain ratioed with the corresponding projection data and the process isrepeated. In each successive iterative repetition, the stored estimatedvolumetric image comes to approach the resultant volumetric image moreclosely.

Backprojection, which is the transpose of the operations of FIG. 4, islikewise computationally efficient since 1D slants from all ringdifferences can be summed prior to a single rotation back to the imagespace. Projection to all segments is additionally speeded through theuse of coarse-depth compression as further described herein.

In one embodiment, a fully-3D rotate and slant projector according tothe present invention is an inherently volumetric projector in that theprojection operation of summing the columns of the re-sampled imagematrix corresponds to integrating the 3D volume contained within thevoxels of the image matrix. The volumetric nature of the LORs isexemplified in FIG. 5. In other embodiments according to the presentinvention, the projector acts as a ray- or line-driven projector andtreats the projection rays as 1D linear elements rather than volumetricelements.

Incorporation/Modeling of Object- and System-Dependent Effects

Embodiments of a rotate-and-slant projector, as described herein,provide a convenient image-based framework for modeling object- andsystem-dependent measurement effects. These effects can be separatedinto three classes: (1) multiplicative effects such as attenuation,uniformity and deadtime, which affect the sensitivity of individualLORS; (2) additive effects such as randoms and scatter, which contributeto the number of prompt coincidences measured but do not provide highresolution spatial information; and (3) geometric effects which map theprecise location of each individual LOR including finite point responsescaused by positron range, non-collinearity effects, anddepth-of-interaction. Incorporation of each of type of effect intoembodiments of the rotate-and-slant projector is similar to the methodused for the 2D rotation-based projector as known to those of ordinaryskill in the art, and details on implementing the geometric arccorrection for oblique LORs are provided below.

LOR Position: The rotate-and-slant projector can incorporate theinterpolation to unevenly-spaced LORS, or “arc correction”, directlyinto the rotator as described in FIGS. 2 and 3. The resampling isperformed during the final shear of the 3-pass method-of-shears rotator.By incorporating the resampling directly into the rotator, a separateinterpolation step for the arc correction is avoided, thus yielding bothless interpolation error and high computational efficiency. This wasdescribed in detail for the 2D case in D. J. Kadrmas, “LOR-OSEM:Statistical PET Reconstruction From Raw Line-of-Response Histograms,”Phys. Med. Biol., vol. 49, pp. 473 1-4744, 2004, which is fullyincorporated by reference herein and made a part hereof, and applicationto embodiments of the 3D projector directly follow the method for the 2Dprojector.

For fully-3D data, there are additional complexities in the exactpositioning of oblique LORs. For example, as shown in FIG. 2, the polarangle for a given ring difference actually changes as a function of sfor a cylindrical scanner. Similarly, “direct plane” and “cross plane”slices are usually stacked by alternating between even and odd ringdifferences to form a final slice thickness that is one-half the ringspacing. This causes the polar angle for oblique LORs at adjacent slicesto alternate in value. In both cases described above, the exact positionand polar angle of each LOR is easily modeled in the slant step of therotate-and-slant projector (making it a spatially-variant 1D operation),providing an exact LOR-based projector, which avoids many of theapproximations commonly used in other PET projectors.

Computational Efficiency and Optimizations

Embodiments of a rotate-and-slant projector can be implemented in Ccomputer programming language as is known in the art, though it is to beappreciated that other programming languages can be utilized and arecontemplated within the scope of the invention. Employed within the codethat comprises projector embodiments are a number of optimizations toreduce the processing time required for projection and reconstruction.Embodiments according to the present invention exploit a number ofsymmetries and other numerical matters to greatly improve thecomputational efficiency of the projector. Computational enhancementscan have their effects upon projection and reconstruction times measuredusing the 3D ordered-subsets expectation-maximization (OSEM) algorithm,as known to one of ordinary skill in the art. Measurements of CPUprocessing times can be performed on a single-CPU such as, for example,a 2.66 GHz Intel Xeon Linux workstation, though it is to be appreciatedthat other CPUs or processors are contemplated within the scope of thisinvention. It is also to be appreciated, however, that embodimentsaccording to the present invention can be implemented usingparallel-implementations on multiple processors as desired.

Coarse-Depth Compression: When using a rotation-based projector, thevolume-integral computations correspond to summing each column of therotated image as shown in relation to FIG. 4. Essentially, each row ofthe rotated image corresponds to a different “depth” between detectorelements (in scanners having a cylindrical symmetry, the “zero” depthcan be considered to fall halfway between the detector elements—i.e., atthe plane intersecting the axis of the detector cylinder—anddepth-dependent effects will often be vertically symmetric about thiszero depth). Projection to oblique segments involves a depth-dependentaxial shear operation (“slant”) which resamples the rotated image to theoblique LORs. Since the oblique LORs are at small angles away fromvertical (θ is small), relatively little depth information is requiredto accurately perform this slant. Embodiments according to the presentinvention incorporate a coarse-depth compression factor into the rotatorwhich collapses the image vertically during the second X shear(reference FIG. 3), essentially compressing the depth information of therotated image. As a result, both memory utilization and the number ofoperations to perform multiple slants to the oblique segments arereduced.

FIGS. 6 a and 6 b are exemplary illustrations of the effect of usingcoarse-depth compression factors upon projection time (FIG. 6 a) andobject resolution in the axial direction (FIG. 6 b), measured as thefull width at half maximum (FWHM) of the 13 mm diameter sphere in theNEMA phantom as imaged on the General Electric Discovery ST™ PET/CTscanner. Here, a large collapse factor effectively eliminates depthinformation from the data, producing axial blurring essentially the sameas for single-slice rebinning (SSRB). It can be seen that projectiontimes are markedly reduced for larger coarse-depth compression factorsas compared to using all the depth information (i.e., coarse compressionfactor=1). FIG. 6 b shows that the axial image resolution was largelyunaffected for coarse-compression factors of about eight and below.

Array Indexing: The orders in which the reconstructed image matrix andprojection data matrix arrays are indexed have an effect upon processingtime. Accounting for the manner in which the arrays are accessed withinthe projector routine, the array indices can be ordered in such a way asto access contiguous blocks of memory when possible. For example, the 3Dreconstructed image matrix would typically be indexed with threeindices: the i,j position within transaxial slice k. This array wouldcommonly be ordered such that i varies fastest, followed by j, and kwould vary the slowest; this ordering is generally referred to as“i:j:k”. In order words, the (i,j,k) element of the image array would beat memory location i+(Xdim*j)+(Xdim*Ydim*k), where Xdim, Ydim, and Zdimare the dimensions of the i, j, and k directions, respectively.

The i:j:k ordering is often efficient for applications which treat theimage one slice at a time, e.g. a transaxial image display program. Forfully-3D projection and reconstruction, however, it is preferable totreat the image volume as a whole. Considering the three steps of therotate-and-slant projector as shown in FIG. 4, several differentordering schemes may be considered. Comparing projection andreconstruction times for three ordering schemes: i:j:k, k:i:j, andk:j:i., it can be seen that the degree of improvement for the differentordering schemes depends upon the image dimensions. The k:i:j orderingscheme consistently provides projection and reconstruction times thatwere about 41-66% as long as for i:j:k ordering, with k:j:i falling inbetween. This is caused at least in part because the innermostprojection loop is a 1D axial interpolation procedure, and the k:i:jindexing allowed contiguous memory blocks to be accessed for thisinterpolation.

Two different projection data indexing orders were also studied indeveloping the embodiments according to the present invention,bin:slice:angle and slice:bin:angle. The slice:bin:angle indexing orderwas consistently found to be about 12-17% faster and is generally apreferable embodiment according to this invention; however it is also tobe appreciated that bin:slice:angle indexing is contemplated within thescope of this invention. Overall, optimizing the image and projectiondata array ordering schemes resulted in almost a factor of two savingsin projection and reconstruction times.

Vertical Projection Symmetry: The approximately cylindrical geometry ofmost modern scanners such as, for example, PET scanners provides whatcan be referred to as vertical symmetry for the projection operation.Referring back to FIG. 4, Step 2 of the rotate-and-slant projectionutilizes a depth-dependent axial slant of the rotated image matrix. Ifthe zero depth is defined to be at the center of the image matrix (i.e.,at the depth of the axis of symmetry of the scanner), then the slant forrows at positive depths is the reverse of the slant for rows at negativedepths. Since the slant requires a row-by-row 1D linear interpolation,the interpolation factors can be shared for the positive and negativedepths. In effect, this replaces two multiplications with two additionsfor each voxel in the row, and this occurs within the innermost loop ofthe slanting routine. This can result in a time savings in the number ofoperations. For example, in one instance this resulted in a time savingsof about 18%.

Other Optimizations: A number of other optimizations were also testedwhich had lesser impact upon the total processing time. For example,once the image matrix has been rotated to angle φ (refer to Step 1 ofFIG. 4), the rotated image at angle+90⁰ can rapidly be obtained byswapping the i- and j-indices of the rotated matrix. This can provide amoderate reduction in processing times, but may be less effective whenthe arc correction (for direct LOR-based reconstruction) is incorporateddirectly into the rotator.

Examples and Evaluation Methods

The following examples are put forth so as to provide those of ordinaryskill in the art with a complete disclosure and description of how thecompounds, compositions, articles, devices and/or methods claimed hereinare made and evaluated, and are intended to be purely exemplary of theinvention and are not intended to limit the scope of what the inventorsregard as their invention. Efforts have been made to ensure accuracywith respect to numbers (e.g., amounts, temperature, etc.), but someerrors and deviations should be accounted for. Unless indicatedotherwise, parts are parts by weight, temperature is in ° C. or is atambient temperature, and pressure is at or near atmospheric.

Implementation of an embodiment of a rotate-and-slant projector forLOR-based reconstruction according to the present invention utilizesknowledge of the LOR positions (including edges of the volumetric LORs)for the PET tomograph of interest, which can be accomplished using datatables for each scanner. The projector can be configured for variousscanners such as, for example, the Biograph Hi-Rez™ scanner (SiemensMedical Solutions), and the Advance™ and Discovery ST™ scanners (GeneralElectric Healthcare), though other scanners are contemplated within thescope of this invention.

Reconstruction Performance: Fully-3D iterative reconstructionperformance with the proposed and conventional projectors were evaluatedusing five reconstruction schemes: (1) Fourier rebinning (FORE) followedby 2D attenuation-weighted OSEM (AW-OSEM) with the ray-driven projector;(2) conventional 3D AW-OSEM using the ray-driven projector, where themeasured projection data were arc-corrected prior to the reconstructionstep; (3) 3D LOR-OSEM using the ray-driven projector; (4) thedistance-driven projector; and (5), an embodiment of therotate-and-slant projector according to the present invention. The 3DLOR-OSEM methods reconstructed directly from the raw PET histograms(i.e., were ordinary Poisson methods), where all corrections includingthe arc-correction were incorporated in the projector/backprojector ofthe iterative reconstruction. In all cases, the exact samescatter+randoms estimates and attenuation normalization factors wereused, though the order of implementation of each differed for schemes(1), (2), and (3)-(5) according to the requirements of each scheme. Forexample, the scatter+randoms estimate was pre-subtracted prior to FOREfor reconstruction scheme (1); it was arc-corrected and then added tothe forward projection of the iterative reconstruction for scheme (2);and it was added (without arc-correction) to the forward projections ofschemes (3)-(5).

The different reconstruction schemes were evaluated and compared usingexperimentally acquired phantom data as described below:

NEMA Phantom Experiment: The NEMA image quality phantom was imaged in 3dmode on a Discover ST™ scanner (GE Healthcare). The phantom was filledwith a total of 3.0 mCi ¹⁸F in water, and contained cold spheres of 37and 28 mm diameter and hot spheres (*:1 target:background) of 17, 13 and10 mm diameter. The phantom was centered in the scanner field-of-viewand images for approximately 7.5 minutes. The raw data and scannernormalization was offloaded to a work station and the data wereprocessed using each reconstruction method under investigation.

Resolution & Contrast Phantom Experiment. The rotate-and-slant projectorwas used to perform fully-3D LOR-OSEM reconstructions of the DeluxeJaszczak Phantom (Data Spectrum Corp., Hillsborough, N.C.). The phantomcontained a plastic insert of six wedges of hot rods with diameters 4.8,6.4, 7.9, 9.5, 11.1, and 12.7 mm, arranged with center-to-center spacingin each wedge equal to twice the rod diameter. Six cold spheres ofdiameters 9.5, 12.7, 15.9, 19.1, 25.4, and 31.8 mm were also placed inthe phantom. The background was filled with 50 MBq ¹⁸F-FDG in water,positioned on the imaging table of the Advance scanner (GE Healthcare)with the rods aligned with the long axis of the scanner, and centered inthe field-of-view. High count data was acquired by scanning the phantomfor 30 minutes in 3D mode with the septa retracted, saving delayedcoincidences in a separate data file for future use. A 10 minutetransmission scan was then acquired using rotating ⁶⁸Ge pin sources. Thescan data, plus all scanner calibrations, normalizations, randoms andscatter estimates were offloaded to a Linux workstation and processedoffline into multiplicative, additive, and geometric components asdescribed above. The data was reconstructed onto 128×128×35 imagematrices with 2.0 mm in-plane pixels and 4.25 mm thick slices using 3DLOROSEM with 6 iterations and 16 subsets.

FIG. 7 shows example images of the Deluxe Jaszczak Phantom,qualitatively demonstrating the image quality that can be obtained withan embodiment of the projector described herein and 3D LOR-OSEM. The 4.8mm rods are clearly resolved on this dataset, and the smallest 9.5 mmcold sphere is likewise clearly resolved. Small circles of radioactivitysurrounding the support posts for the spheres can also be seen betweenthe wedges of hot rods. The images for the rotate-and-slant projectorand the other reconstruction methods show similar image quality, and themost significant differences noted were differences in reconstructednoise texture.

Analysis Methods: The reconstruction methods identified above werecomparatively evaluated in terms of reconstructed image spatialresolution, hot-object contrast, and background noise, taking intoconsideration differences in the rate of iterative convergence resultingfrom the different LOR models inherent in different projectors. Theeffect of using coarse-depth compression upon axial resolution wasanalyzed by fitting Gaussian distributions to axial profiles of the 13mm diameter hot sphere from the NEMA phantom experiment. The full-widthat half-maximum (FWHM) of the fitted Gaussians were analyzed as afunction of coarse-depth compression factor, and compared to those forreconstruction using SSRB.

Three analysis measures were computed for the Deluxe Jaszczak phantomexperiment: (1) the average peak/valley ratio for the wedge of 4.8 mmhot rods, which is a measure closely-related to the recovered spatialresolution; (2) the contrast of the 12.7 mm diameter cold sphere wascomputed as (FG-BG)/(FG+BG), yielding a figure-of-merit where 0.0 and1.0 reflect no contrast and perfect contrast, respectively; and (3) thestandard deviation of background voxels, quoted as a percent of the meanvalue for a large (256 voxel) background region of uniform activityconcentration, was computed as a measure of statistical noise in theimage. These three measures were first computed as a function ofiteration, and then the noise measure was analyzed as a function of theresolution and contrast measures to account for differences in the rateof iterative recovery of image features for each of the reconstructionmethods.

Projection and Reconstruction Times: In one instance, the CPU timerequired for fully-3D projection and 3D LOR-OSEM reconstruction weremeasured using datasets representative of several PET tomographs,including the Advance™ (GE, 283 LORs×336 angles, 35 slices), DiscoveryST™ (GE, 249 LORs×210 angles, 47 slices), and Hi-Rez Biograph™ (SiemensMedical Solutions, 335 LORs×336 angles, 81 slices). The projection andreconstruction times were measured as a function of the number ofoblique angles included in the reconstruction. For example, using amaximum ring difference of 25 and a span of 3, 17 sets of obliquesinograms would be obtained: (0, ±1), (+2, +3, +4), (−2, −3, −4), . . ., (−23, −24, −25). FIGS. 8 a and 8 b show the full projection anditerative reconstruction times as a function of the number of obliquedatasets included. The points on these curves, which are representativeof common operating points for the three scanners, are marked. Theprojection and reconstruction times for 2D data using the rotation-basedprojector (same C programming code with same optimizations) are alsomarked on the figure.

It can be seen from FIGS. 8 a and 8 b that the projection time for anembodiment of a rotate-and-slant projector according to the presentinvention includes a component associated with 2D projection andincreases from there linearly with the number of oblique segments. Incomparison, the projection time for other fully-3D projectors, such asline- or ray-driven projectors, typically scales linearly with the totalnumber of projection rays (LORs) because each oblique segment addssignificant time to the computation. FIG. 8 a shows the projection CPUtimes and FIG. 8 b shows the full reconstruction times for fouriterations of a fully-3D LOR-OSEM with 14 subsets for two modern OETscanners, plotted as a function of the total number of oblique segmentsincluded. The horizontal axes cover the range from 2D to fully-3D cases.Processing time for the conventional projectors scales poorly withincreasing numbers of oblique segments, whereas an embodiment of therotate and slant projector according to the present invention scalesefficiently, even for the complete fully-3D cases. This is because ofthe manner in which embodiments of the rotate and slant projectorefficiently conserve azimuthal operations for computing 3D projectionsto multiple oblique segments. Note that in FIG. 8 b, the fully-3DLOR-OSEM with the rotate and slant projector is nearly as fast asrebinning followed by 2D reconstruction (FORE AW-OSEM2D).

FIG. 9 shows how the projection time for the rotate-and-slant projectorincreases as a function of transaxial (transverse plane) image matrixsize. In FIG. 9, all the times are for projection of a 47 slice imagesto 249 LORs×210 angles×47 slices by 23 ring differences. The projectiontime scales roughly with the square of the image matrix dimension, andremains reasonable on modern computers for matrices as large asapproximately 256×256. Larger matrix sizes are included in FIG. 9 toprovide a rough approximation of how the projector would scale for usewith other modalities such as, for example, x-ray CT.

Projection Accuracy and Reconstructed Image Quality: Results of aquantitative analysis of image quality are shown in FIGS. 10 a, 10 b, 10c, and 10 d. Three image quality measures were studied for the DeluxeJaszczak phantom experiment described above: (1) the average peak/valleyratio for the 6.4 mm hot rods, providing a measure closely related tospatial resolution as shown in FIG. 10 a; (2) the contrast ratio of the12.7 mm cold sphere as shown in FIG. 10 b; and (3) the averagebackground noise, calculated as the percent standard deviation versusthe mean of the background as shown in FIGS. 10 c and 10 d. FIGS. 10 and10 b demonstrate differences in the rates of iterative recovery of theseimage features, which arise because of differences in the projectionmodels for each of the methods. FIGS. 10 c and 10 d effectivelynormalize for this effect, permitting comparison of image noise ascontrast resolution or contrast. The data demonstrate a trend towardimproved image quality measures when moving from rebinning to 3D AW-OSEMto 3D LOR-OSEM. This reflects the improved statistical models of thefully-3D iterative methods, coupled with reduced image degradation whenarc correction is included in the projector (LOR-based) as opposed topre-correcting for this effect (AW-OSEM).

Methods of Use

FIG. 11 is a flow chart illustrating an exemplary process for iterativefully-3D reconstruction of an image by positron emission tomography(PET) by a scanner having a plurality of detector elements. At step1100, a resultant image volume is estimated. The estimated image volumeis stored in an image matrix in a memory. At step 1102, the image matrixis rotated into alignment with an azimuthal angle of a projection ray orline-of-response between detector elements in a same transaxial plane ofthe scanner. The projection ray or line-of-response is perpendicular toan axis of symmetry of the scanner. At step 1104, the image volume isre-sampled at the azimuthal angle by slanting the image matrix to eachaxial angle of oblique projection rays or lines of response at thatazimuthal angle. The oblique projection ray or line-of-response isbetween detector elements in different transaxial planes of the scannersuch that columns of the image matrix align with a plurality of datacollection directions. At step 1106, the re-sampled image is projectedby summing the columns of the image matrix. In some instances this maybe accomplished by integrating a 3D volume contained within avolume-of-overlap. With volumetric integration, the image matrixrotation and slanting as described in steps 1102 and 1104, above, areperformed using the volume-of-overlap between 3D volumes of image matrixvoxels and a 3D volume of the projection rays or lines-of-response andthe oblique projection rays or lines-of-response.

In other embodiments according to the present invention, a coarse-depthcompression factor is applied to the image matrix such that rows of theimage matrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed, and the image is projected by summing thecompressed columns of the image matrix.

CONCLUSION

While this invention has been described in connection with preferredembodiments and specific examples, it is not intended that the scope ofthe invention be limited to the particular embodiments set forth, as theembodiments herein are intended in all respects to be illustrativerather than restrictive.

Unless otherwise expressly stated, it is in no way intended that anymethod set forth herein be construed as requiring that its steps beperformed in a specific order. Accordingly, where a method claim doesnot actually recite an order to be followed by its steps or it is nototherwise specifically stated in the claims or descriptions that thesteps are to be limited to a specific order, it is no way intended thatan order be inferred, in any respect. This holds for any possiblenon-express basis for interpretation, including: matters of logic withrespect to arrangement of steps or operational flow; plain meaningderived from grammatical organization or punctuation; the number or typeof embodiments described in the specification.

Throughout this application, various publications are referenced. Thedisclosures of these publications in their entireties are herebyincorporated by reference into this application in order to more fullydescribe the state of the art to which this invention pertains.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the scope or spirit of the invention. Otherembodiments of the invention will be apparent to those skilled in theart from consideration of the specification and practice of theinvention disclosed herein. It is intended that the specification andexamples be considered as exemplary only, with a true scope and spiritof the invention being indicated by the following claims.

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1. A projector for iterative fully-3D reconstruction of an image bypositron emission tomography (PET) using a scanner comprised of aplurality of detector elements comprising: a volume image memorycontaining a resultant image volume that is initially estimated andstored in an image matrix; a projection matrix, wherein said projectionmatrix is comprised of planes of projection data collected at each of aplurality of data collection directions and said data collectiondirections include azimuthal angles of projection rays orlines-of-response that are perpendicular to the axis of symmetry of thescanner between detector elements in a same transaxial plane of thescanner and axial angles of oblique projection rays or lines of responsebetween detector elements in different transaxial planes of the scanner;and a rotator that re-samples said resultant image volume from saidvolume image memory by rotating the image matrix to a given azimuthalangle of the projection matrix and slanting the columns of the imagematrix to each of the axial angles of oblique projection rays orlines-of-response at that given azimuthal angle such that columns of theimage matrix align with each of the plurality of data collectiondirections of the projection matrix, wherein said rotator applies acoarse-depth compression factor to said image matrix such that rows ofthe image matrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed and said image is projected by summing thecompressed columns of the image matrix.
 2. The projector of claim 1,wherein memory utilization and the number of operations to performmultiple slants to the oblique projection rays or lines-of-response arereduced by application of said coarse-depth compression factor to saidimage matrix.
 3. The projector of claim 1, wherein said scanner iscomprised of a plurality of substantially adjacent rings of detectorelements in its axial direction.
 4. The projector of claim 1, whereinsaid scanner is a PET scanner and said PET scanner or software used toprocess data from said PET scanner includes said projector.
 5. Aprojector for iterative fully-3D reconstruction of an image by positronemission tomography (PET) using a scanner comprised of a plurality ofdetector elements comprising: a volume image memory containing aresultant image volume that is initially estimated and stored in animage matrix; a projection matrix, wherein said projection matrix iscomprised of planes of projection data collected at each of a pluralityof data collection directions and said data collection directionsinclude azimuthal angles of projection rays or lines-of-response thatare perpendicular to the axis of symmetry of the scanner betweendetector elements in a same transaxial plane of the scanner and axialangles of oblique projection rays or lines of response between detectorelements in different transaxial planes of the scanner; and a rotatorthat re-samples said resultant image volume from said volume imagememory by rotating the image matrix to a given azimuthal angle of theprojection matrix and slanting the columns of the image matrix to eachof the axial angles of oblique projection rays or lines of response atthat given azimuthal angle such that columns of the image matrix alignwith each of the plurality of data collection directions of theprojection matrix, wherein said image matrix rotation and slanting isperformed using volume-of-overlap between 3D volumes of image matrixvoxels and a 3D volume of said projection rays or lines-of-response andsaid oblique projections rays or lines-of-response, such that projectionby summing the columns of the image matrix integrates the 3D volumecontained within said volume-of-overlap.
 6. The projector of claim 5,wherein said scanner is comprised of a plurality of substantiallyadjacent rings of detector elements in its axial direction.
 7. Theprojector of claim 5, wherein said scanner is a PET scanner and said PETscanner or software used to process data from said PET scanner includessaid projector.
 8. A projector for iterative fully-3D reconstruction ofan image by positron emission tomography (PET) using a scanner comprisedof a plurality of detector elements comprising: a volume image memorycontaining a resultant image volume that is initially estimated andstored in an image matrix; a projection matrix, wherein said projectionmatrix is comprised of planes of projection data collected at each of aplurality of data collection directions and said data collectiondirections include azimuthal angles of projection rays orlines-of-response that are perpendicular to the axis of symmetry of thescanner between detector elements in a same transaxial plane of thescanner and axial angles of oblique projection rays or lines of responsebetween detector elements in different transaxial planes of the scanner;and a rotator that re-samples said resultant image volume from saidvolume image memory by rotating the image matrix to a given azimuthalangle of the projection matrix and slanting the columns of the imagematrix to each of the axial angles of oblique projection rays orlines-of-response at that given azimuthal angle such that columns of theimage matrix align with each of the plurality of data collectiondirections of the projection matrix, wherein said rotator applies acoarse-depth compression factor to said image matrix such that rows ofthe image matrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed and wherein said image matrix rotation andslanting is performed using volume-of-overlap between 3D volumes ofimage matrix voxels and a 3D volume of said projection rays orlines-of-response and said oblique projection rays andlines-of-response, such that projection by summing the columns of theimage matrix integrates the 3D volume contained within saidvolume-of-overlap.
 9. The projector or claim 8, wherein memoryutilization and the number of operations to perform multiple slants tothe oblique projection rays or lines-of-response are reduced byapplication of said coarse-depth compression factor to said imagematrix.
 10. The projector of claim 8, wherein said scanner is comprisedof a plurality of substantially adjacent rings of detector elements inits axial direction.
 11. The projector of claim 8, wherein said scanneris a PET scanner and said PET scanner or software used to process datafrom said PET scanner includes said projector.
 12. A method of iterativefully-3D reconstruction of an image by positron emission tomography(PET) by a scanner having a plurality of detector elements comprising:estimating a resultant image volume, wherein said image volume is storedin an image matrix; rotating the image matrix into alignment with anazimuthal angle of a projection ray or line-of-response between detectorelements in a same transaxial plane of the scanner, wherein saidprojection ray or line-of-response is perpendicular to an axis ofsymmetry of the scanner; re-sampling said image volume at said azimuthalangle by slanting said image matrix to each axial angle of obliqueprojection rays or lines of response at that azimuthal angle, whereineach said oblique projection ray or line-of-response is between detectorelements in different transaxial planes of the scanner such that columnsof the image matrix align with a plurality of data collectiondirections; applying a coarse-depth compression factor to said imagematrix such that rows of the image matrix are collapsed onto one anotherin accordance with the coarse-depth compression factor so depthinformation of the rotated image matrix is compressed; and projectingsaid image by summing the compressed columns of the image matrix. 13.The method of claim 12, wherein said scanner is comprised of a pluralityof substantially adjacent rings of detector elements in its axialdirection.
 14. The method of claim 12, wherein said scanner is a PETscanner and said PET scanner or software used to process data from saidPET scanner includes said projector.
 15. A method of iterative fully-3Dreconstruction of an image by positron emission tomography (PET) by ascanner having a plurality of detector elements comprising: estimating aresultant image volume, wherein said image volume is stored in an imagematrix; rotating the image matrix into alignment with an azimuthal angleof a projection ray or line-of-response between detector elements in asame transaxial plane of the scanner, wherein said projection ray orline-of-response is perpendicular to an axis of symmetry of the scanner;re-sampling said image volume at said azimuthal angle by slanting saidimage matrix to each axial angle of oblique projection rays orlines-of-response at that azimuthal angle, wherein each said obliqueprojection ray or line-of-response is between detector elements indifferent transaxial planes of the scanner such that columns of theimage matrix align with a plurality of data collection directions; andprojecting said image by summing the columns of the image matrix byintegrating a 3D volume contained within a volume-of-overlap, whereinsaid image matrix rotation and slanting is performed using thevolume-of-overlap between 3D volumes of image matrix voxels and a 3Dvolume of said projection rays or lines-of-response and said obliqueprojection rays or lines-of-response.
 16. A method of iterative fully-3Dreconstruction of an image by positron emission tomography (PET) by ascanner having a plurality of detector elements comprising: estimating aresultant image volume, wherein said image volume is stored in an imagematrix; rotating the image matrix into alignment with an azimuthal angleof a projection ray or line-of-response between detector elements in asame transaxial plane of the scanner, wherein said projection ray orline-of-response is perpendicular to an axis of symmetry of the scanner;re-sampling said image volume at said azimuthal angle by slanting saidimage matrix to each axial angle of oblique projection rays orlines-of-response at that azimuthal angle, wherein each said obliqueprojection ray or line-of-response is between detector elements indifferent transaxial planes of the scanner such that columns of theimage matrix align with a plurality of data collection directions;applying a coarse-depth compression factor to said image matrix suchthat rows of the image matrix are collapsed onto one another inaccordance with the coarse-depth compression factor so depth informationof the rotated image matrix is compressed; and projecting said image bysumming the columns of the image matrix by integrating a 3D volumecontained within a volume-of-overlap, wherein said image matrix rotationand slanting is performed using the volume-of-overlap between 3D volumesof image matrix voxels and a 3D volume of said projection rays orlines-of-response and said oblique projection rays andlines-of-response.
 17. The method of claim 16, wherein said scanner iscomprised of a plurality of substantially adjacent rings of detectorelements in its axial direction.
 18. The method of claim 16, whereinsaid scanner is a PET scanner and said PET scanner or software used toprocess data from said PET scanner includes said projector.
 19. Apositron emission tomography (PET) scanner comprising: a plurality ofdetector elements; and a rotate and slant projector comprised of aprojection matrix, an image matrix and a rotator, wherein the rotatorre-samples a resultant image volume from a volume image memory byrotating the image matrix to a given azimuthal angle of the projectionmatrix and slanting columns of the image matrix to each axial angle ofoblique projection rays or lines-of-response at that given azimuthalangle such that columns of the image matrix align with each of theplurality of data collection directions of the projection matrix;wherein said rotator applies a coarse-depth compression factor to saidimage matrix such that rows of the image matrix are collapsed onto oneanother in accordance with the coarse-depth compression factor so depthinformation of the rotated image matrix is compressed and said image isprojected by summing the compressed columns of the image matrix.
 20. Apositron emission tomography (PET) scanner comprising: a plurality ofdetector elements; and a rotate and slant projector comprised of aprojection matrix, an image matrix and a rotator, wherein the rotatorre-samples a resultant image volume from a volume image memory byrotating the image matrix to a given azimuthal angle of the projectionmatrix and slanting columns of the image matrix to each axial angle ofoblique projection rays or lines-of-response at that given azimuthalangle such that columns of the image matrix align with each of theplurality of data collection directions of the projection matrix;wherein said rotator performs said image matrix rotation and slantingusing volume-of-overlap between 3D volumes of image matrix voxels and a3D volume of said projection rays or lines-of-response and said obliqueprojection rays or lines-of-response, such that projection by summingcolumns of the image matrix integrates the 3D volume contained withinsaid volume-of-overlap.
 21. A positron emission tomography (PET) scannercomprising: a plurality of detector elements; and a rotate and slantprojector comprised of a projection matrix, an image matrix and arotator, wherein the rotator re-samples a resultant image volume from avolume image memory by rotating the image matrix to a given azimuthalangle of the projection matrix and slanting columns of the image matrixto each axial angle of oblique projection rays or lines-of-response atthat given azimuthal angle such that columns of the image matrix alignwith each of the plurality of data collection directions of theprojection matrix; wherein said rotator applies a coarse-depthcompression factor to said image matrix such that rows of the imagematrix are collapsed onto one another in accordance with thecoarse-depth compression factor so depth information of the rotatedimage matrix is compressed and wherein said image matrix rotation andslanting is performed by said rotator using volume-of-overlap between 3Dvolumes of image matrix voxels and a 3D volume of said projection raysor lines-of-response and said oblique projection rays andlines-of-response, such that projection by summing the columns of theimage matrix integrates the 3D volume contained within saidvolume-of-overlap.